Symmetries in exceptional holonomy problems

• Grant number: 23/12372-1
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Effective date (Start): May 01, 2025
• Effective date (End): March 31, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Henrique Nogueira de Sá Earp
• Grantee: Udhav Fowdar
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 21/04065-6 - BRIDGES: Brazil-France interplays in Gauge Theory, extremal structures and stability, AP.TEM

Abstract

Calabi-Yau, G2- and Spin(7)-manifolds are all special classes of Einstein manifolds occurring in Berger's classification of Riemannian holonomy groups. They are also of particular interest in theoretical physics, especially in the context of supersymmetry and the SYZ conjecture for mirror symmetry. In this project we aim to study various problems pertaining to these manifolds. Our main objectives are as follows: (1) The SYZ conjecture suggests that 'mirrors' to special Lagrangian submanifolds in Calabi-Yau 3-folds correspond to certain special connections called deformed Hermitian Yang-Mills (dHYM) connections. Likewise, G2 and Spin(7) analogues of the SYZ conjecture lead to the notion of deformed G2- and deformed Spin(7)-instantons as 'mirrors' to (co)-associatives and Cayley submanifolds. Thus, this suggests an intricate link between the calibrated geometry and gauge theory of mirror pairs. While plenty of dHYM connections are now known, very little is known in the G2 and Spin(7) cases. We aim to construct new examples of these deformed instantons and study their properties, in particular, their deformation theory and moduli spaces. (2) An interesting class of equations originating from supersymmetry are the Hull-Strominger/heterotic systems, which intertwine the torsion of the underlying geometric structure with the curvature of certain special connections. We aim to find new solutions to these heterotic systems on manifolds with large symmetry group, such as homogeneous spaces and cohomogeneity one manifolds. (3) Geometric flows provide a key tool in the search of special geometric structures. We aim to formulate, and hopefully classify, all geometric flows for Sp(n)-structures i.e. almost hyper-Hermitian structures. Of special interest is the Sp(2) case due to its relation with Calabi-Yau 4-folds and Spin(7) manifolds.